JeongHyeong Park (Sungkyunkwan University) – Curvature identities and their applications

Is there a curvature identity that holds on any Riemannian manifold? Through the Chern-Gauss-Bonnet theorem, we can derive curvature identities that apply to 4-dimensional or 6-dimensional Riemannian manifolds. As an application of curvature identities, we prove Lichnerowicz’s conjecture in 4 dimensions under a slightly more general setting. Furthermore, we explore weakly Einstein manifolds, which arise as a generalization of 4-dimensional Einstein manifolds through the application of curvature identities. We also investigate the existence and non-existence of weakly Einstein metrics on certain Lie groups in recent studies, and propose a conjecture based on these results. (This is joint work with Y. Euh, S. Kim and Nikolayevsky.)